Resonance frequency of column (Monolithology)
A column of elastic material (singlelithology) resting on a rigid base has natural periods of vibration that depend on the mode of oscillation. The theoretical value for natural period of oscillation (T) is given by:
where:
 H = column height of soil (for this verification H=8.0 m)
 Vs = velocity of shear waves (for this verification Vs=100 m/s)
 T = natural period of oscillation (for this verification T=0.32 s > F=3.125 Hz)
Geometry and Material Properties
The model is a column whit dimension 8.0x1.0 meters and it make of eight (8) regular elements and eighteen (18) nodes.
Material Properties
Density  Wave Vs  Shear Modulus  Poisson Ratio  Damping 

10 N/m^{2}  100 m/s  1.019E4 N/m^{2}  0.10  1.0 % 
Boudary Conditions
The column have a behavior in shear and the vertical motion (Y) is inhibited to eliminate bending modes. The loading is applied to the base (XY).
You apply the load horizontal component only in the nodes with XDirection locked:
 Node 1, 1 > Fix XYDirection
 Node 1, 2 > Fix XYDirection
Input Motion
The horizontal acceleration (max 0.2g) given by:
where:
 [α] = 2.500
 [β] = 0.011
 [γ] = 8.000
 [ω] (angular frequency) = 2πf
 f = Variable Frequency (1.000, 2.000, 3.000, 4.000, 5.000, 6.000 Hz)
Results
It is shown the maximum amplification obtained when the frequency of the input motion is near to the resonance frequency of the column. The figures show the horizontal acceleration at the top of the column (node 1,9) as a function of time and the acceleration spectrum with a predominant period of 0.325 sec (f = 3.077 Hz).

 Input Acceleration Spectrum

 Output Acceleration Spectrum
Amplification Factor
Frequency Input Motions  1.0 Hz  2.0 Hz  3.0 Hz  4.0 Hz  5.0 Hz  6.0 Hz 

Output FHv [0.1  2.50 Sec]  1.136  1.873  20.530  2.302  1.215  1.003 
Output FHa [0.1  0.50 Sec]  1.144  1.919  20.963  2.305  1.213  1.004 